Why category theory is interesting

This is very preliminary!

Consider the following table:

Arbitrary categories	pullbacks: limits of cospans in arbitrary categories	limits of arbitrary diagrams in arbitrary categories
Category of sets	pullbacks: limits of cospans in the category of sets	limits of arbitrary diagrams in the category of sets
	pullbacks: limits of cospans	limits of arbitrary diagrams

It illustrates how category theory facilitates the process of generalization.
The concept of pullback can be introduced by
a very simple example in the category of sets,
the pullback of a cospan

PolPref: Men --------> Parties <-------- Women :PolPref

discussed here.
This simple example is generalized to
the definition of arbitrary pullbacks in the category of sets,
as shown in the lower left entry of the above table.
That concept may be further generalized in either of two ways,
by generalizing from pullback to arbitrary limits
or from the category of sets to arbitrary categories,
shown by moving to the right or up in the above table.
Finally, one may combine both generalizations as shown in the top right entry.

Still further generalizations are possible and useful,
by considering more general concepts of category,
in most of which the concept of limit is still useful.